If 3y-y^(2)-x^(3)=-y^(3) then find (dy)/(dx) in terms of x and y. Answer: (dy)/(dx)=

Given Equation Differentiation: We are given the equation 3y - y^2 - x^3 = -y^3. To find (dy)/(dx), we need to differentiate both sides of the equation with respect to x, treating y as a function of x (implicit differentiation).Left Side Differentiation: Differentiate the left side of the equation with respect to x: d/dx(3y - y^2 - x^3). This gives us d/dx(3y) - d/dx(y^2) - d/dx(x^3).Differentiate Each Term: Differentiate each term: d/dx(3y) = 3(dy/dx), d/dx(y^2) = 2y(dy/dx) using the chain rule, and d/dx(x^3) = 3x^2.Right Side Differentiation: Differentiate the right side of the equation with respect to x: d/dx(-y^3). This gives us -3y^2(dy/dx) using the chain rule.Combine Differentiated Terms: Combine the differentiated terms to form an equation: 3(dy/dx) - 2y(dy/dx) - 3x^2 = -3y^2(dy/dx).Rearrange to Solve: Rearrange the terms to solve for (dy/dx): (3 - 2y - 3y^2)(dy/dx) = 3x^2.Factor Out (dy/dx): Factor out (dy/dx) from the left side of the equation: (dy/dx)(3 - 2y - 3y^2) = 3x^2.Isolate (dy/dx): Divide both sides by (3 - 2y - 3y^2) to isolate (dy/dx): (dy/dx) = 3x^2 / (3 - 2y - 3y^2).

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If
3y-y^(2)-x^(3)=-y^(3) then find
(dy)/(dx) in terms of
x and
y.
Answer:
(dy)/(dx)=

If $3 y-y^{2}-x^{3}=-y^{3}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

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Algebra 1

Transformations of quadratic functions

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Q. If

3 y-y^{2}-x^{3}=-y^{3}

then find

\frac{d y}{d x}

in terms of

x

and

y

\newline

Answer:

\frac{d y}{d x}=

Given Equation Differentiation: We are given the equation $3y - y^2 - x^3 = -y^3$ . To find $\frac{dy}{dx}$ , we need to differentiate both sides of the equation with respect to $x$ , treating $y$ as a function of $x$ (implicit differentiation).
Left Side Differentiation: Differentiate the left side of the equation with respect to $x$ : $\frac{d}{dx}(3y - y^2 - x^3)$ . This gives us $\frac{d}{dx}(3y) - \frac{d}{dx}(y^2) - \frac{d}{dx}(x^3)$ .
Differentiate Each Term: Differentiate each term: $\frac{d}{dx}(3y) = 3\frac{dy}{dx}$ , $\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}$ using the chain rule, and $\frac{d}{dx}(x^3) = 3x^2$ .
Right Side Differentiation: Differentiate the right side of the equation with respect to $x$ : $\frac{d}{dx}(-y^3)$ . This gives us $-3y^2\frac{dy}{dx}$ using the chain rule.
Combine Differentiated Terms: Combine the differentiated terms to form an equation: $3\left(\frac{dy}{dx}\right) - 2y\left(\frac{dy}{dx}\right) - 3x^2 = -3y^2\left(\frac{dy}{dx}\right)$ .
Rearrange to Solve: Rearrange the terms to solve for $(\frac{dy}{dx})$ : $(3 - 2y - 3y^2)(\frac{dy}{dx}) = 3x^2$ .
Factor Out $(\frac{dy}{dx}):$ Factor out $(\frac{dy}{dx})$ from the left side of the equation: $(\frac{dy}{dx})(3 - 2y - 3y^2) = 3x^2$ .
Isolate $\frac{dy}{dx}$ : Divide both sides by $(3 - 2y - 3y^2)$ to isolate $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{3x^2}{3 - 2y - 3y^2}$ .